Unimodular Lattice
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Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. In mathematics, a unimodular lattice is a lattice of determinant 1 or 1. The E8 lattice and the Leech lattice are two famous examples. This is given by all vectors (a1,...,am+n) in Rm,n such that either all the ai are integers or they are all integers plus 1/2, and their sum is even. The lattice II8,0 is the same as the E8 lattice. Positive definite unimodular lattices have been classified up to dimension 25. There is a unique example In,0 in each dimension n less tha...
Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. In mathematics, a unimodular lattice is a lattice of determinant 1 or 1. The E8 lattice and the Leech lattice are two famous examples. This is given by all vectors (a1,...,am+n) in Rm,n such that either all the ai are integers or they are all integers plus 1/2, and their sum is even. The lattice II8,0 is the same as the E8 lattice. Positive definite unimodular lattices have been classified up to dimension 25. There is a unique example In,0 in each dimension n less than 8, and two examples (I8,0 and II8,0) in dimension 8. The number of lattices increases moderately up to dimension 25 (where there are 665 of them), but beyond dimension 25 the Smith-Minkowski-Siegel mass formula implies that the number increases very rapidly with the dimension; for example, there are more than 80,000,000,000,000,000 in dimension 32.
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